We study the problem of achieving integral input to state stability (iISS) with respect to noise for a class of upper triangular nonlinear systems with uncertainty and measurement noise. We propose a novel step-by-step Lyapunov-based design, consisting of (1) splitting an n-dimensional system into n one-dimensional systems, each with its own state, inputs, and measurement, (2) constructing a one-dimensional measurement feedback controller for each one-dimensional system, according to a certainty equivalence principle, and (3) selecting the parameters of these controllers so that their interconnection gives a measurement feedback controller for the n-dimensional system. The stability analysis is performed through filtered Lyapunov functions, which are Lyapunov functions with parameters being the output of suitable dynamical filters.